Chris Colefax wrote:
> 
> I don't believe it is - from my studies of splines a while back it
> seems the integration of the relevant functions just isn't possible.

A quadratic spline

  P(x) = Ax^2 + Bx + C

has the derivative in x of

  dP(x)/dx = 2Ax + B

which has a magnitude of

  sqrt( 4(A.A)x^2 + 4(A.B)x + (B.B) ),

which can be integrated over x (the integral is in my calc book).

A cubic spline,

  P(x) = Ax^3 + Bx^2 + CX + D

Will have a quadratic derivative in x, namely

  dP/dx = 3Ax^2 + 2Bx + C

The magnitude of which is

  |dP/dx| = sqrt( 9(A.A)x^4  + 12(A.B)x^3 +  4(B.B)x^2 +  6(A.C)x^2
     +  4(B.C)x +   (C.C) )

Dunno if this can be integrated over x.  Taylor's expansion might
provide a clue to the integral.

Regards,
John
-- 
Rusty is rendering!

Post a reply to this message